Inferring Synaptic Update Rules in a Neural Simulator Honours Thesis Ryan Fayyazi April 2020
HMM for static neural circuit Synaptic Weight Matrix Membrane Potential Intracellular [Ca 2+ ] Calcium Fluorescence
Νeuroscience Background Membrane Potential ● determines neuron’s activity (depolarization = active, hyperpolarization = suppressed) ● membrane potential = electrical potential inside neuron - electrical potential outside neuron electrical potentials determined by concentrations of charged ions (e.g. Na + , K + , Cl - ) ●
Νeuroscience Background Intracellular [Ca 2+ ] ● increases when neuron’s membrane is depolarized Calcium Fluorescence measures intracellular [Ca 2+ ] using molecules ● which fluoresce when they bind calcium ● indirect measure of neuron activity
Synaptic Plasticity Synaptic Weight Matrix ● Synapse: junction where membrane potential of one neuron influences membrane potential of another electrical synapse chemical synapse ● Synaptic Weight: abstraction denoting influence exerted by one neuron on the other ○ Synaptic weight determined by receptor, channel, presynaptic vesicle density, etc.
Deterministic Simulator
Synaptic Plasticity Donald Hebb (1949): When cell A “repeatedly or persistently” takes part in firing cell B, the efficiency of A’s signal to B (weight of synapse) is increased by some physiological process Cell A Cell B Bliss & Lomo, 1973 Most learning theories incorporate the idea that synaptic plasticity is a fundamental mechanism by which behavioural response is modified.
Synaptic Plasticity ?
Formalizing Synaptic Update Rules Donald Hebb (1949): When cell A “repeatedly or persistently” takes part in firing cell B, the efficiency of A’s signal to B (weight of synapse) is increased by some physiological process Compositional structure: upstream synaptic weights S = div({}, mul({}, {})) presynaptic firing rates postsynaptic firing rate Continuous parameter(s): rate constant 𝛊 =
SURF Goal ? Model synaptic weight dynamics underlying plasticity in a given behaviour Big picture: infer given observations of neural activity in circuit underlying behaviour, during learning
Simplifying Assumptions ? 1) Deterministic simulator and : ● only need and ● maximum a posteriori estimate is decent approximation of 2) Finite set of candidate structures : ● first step towards difficult search over infinite discrete structure space
Continuous Optimization ? Objective
Continuous Optimization ? Objective
Continuous Optimization ? Objective Rayleigh distribution with scale parameter for each non-zero entry equaling the experimentally determined naive weights
Continuous Optimization ? Objective
Continuous Optimization ? Objective By LLN Approximate expectation with Monte Carlo integration
Continuous Optimization ? Objective This MC integration requires: 1) Samples ● Sample with simulator, initialize randomly 2) Emission density
Synaptic Update Rule Finder
Experiment Plastic behaviour of interest: Tap-withdrawal response habituation in C. elegans (roundworm) Andrew Giles Giles & Rankin, 2009
Experiment Tap-withdrawal circuit has been identified ● Mechanosensory neuron-interneuron synapses thought to be site of plasticity ● Simulator = circuit + ODEs
Experiment Problem: No available observations from tap-withdrawal circuit during habituation Solution: Build synthetic observations using Hebb’s rule, which results in habituation-ish simulator dynamics 1) Initialize v o and c o independently with samples from Gaussian (c) (e) with experimentally determined naive synaptic weights 2) Initialize w o and w 0 Set R (c) and R (e) to Hebb’s rule, τ w (c) = τ w (e) = 0.001 3) 4) Simulate forward with habituation-inducing currents injected into mechanosensory neurons 5) Sample calcium fluorescence observations with
Results: Generated Candidate Rules Observation-generating rule pair: Observation-generating rules: Candidate rule pair A Sampled with recursive random rule generator G(d) Candidate rule pair B Same structure as “true” rules Candidate rule pair C
Results Candidate pair with “true” structure achieved lowest loss 0 -1000 -2000 Loss (Not Normalized) -3000 -4000 -5000 -6000 -7000 3800 4000 4200 4400 4600 4800 5000 Optimization Step
Results Candidate pair with “true” structure produced best qualitative reconstruction of latent dynamics during habituation after optimization Candidate rule pair A Candidate rule pair B Candidate rule pair C
Results (c) and w 0 (e) . Candidate pair with “true” structure achieved correct initial synaptic weights w 0 Electrical Synapses Chemical Synapses
Future Directions Develop strategy for searching over infinite, discrete space of rule structures ● This thesis showed that given enough samples, SURF finds the correct rule and initial weights ● Frame as infinitely many-armed bandit with finite gradient descent budget during structure exploration Test SURF using observations (1) which capture more of habituation’s characteristic features, and (2) from real organisms undergoing habituation ● Infer intracellular [Ca2+] in tap-withdrawal neurons from videos of worms undergoing habituation, and convert this inferred value to fluorescence ● Use feature-based optimization to estimate rule and initial weights from behavioural data (e.g. reversal magnitude) gathered during habituation Perform bayesian inference instead of maximum a posteriori estimation ● Use stochastic simulator ● Perform inference with sequential Monte Carlo or Metropolis-Hastings estimation.
References Adriel, E. L., & Rankin, C. H. (2010). An elegant mind: Learning and memory in Caenorhabditis elegans. Learning & Memory, 17 (4), 191-201. Bliss, T. V. P., & Lømo, T. (1973). Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path. Journal of Physiology, 232 , 331-356. Dayan P., & Abbott, L. F. (2001). Theoretical Neuroscience: Computational and mathematical modeling of neural systems. The MIT Press, Cambridge, MA. Dudek, S. M., & Bear, M. F. (1992). Homosynaptic long-term depression in area CA1 of hippocampus and effects of n-methyl-d-aspartate receptor blockade. Proceedings of the National Academy of Sciences of the United States of America, 89, 4363-4367 . Giles, A. C., & Rankin, C. H. (2008). Behavioral and genetic characterization of habituation using Caenorhabditis elegans. Neurobiology of Learning and Memory, 92, 139-146. Hebb, D. O. (1949). The Organization of Behaviour: A Neuropsychological Theory. Wiley, New York, New York. Kandel, E. (2013). Principles of Neural Science, Fifth Edition . The McGraw-Hill Companies. Purves, D., Augustine, G. J., Fitzpatrick, D., Hall, W. C., LaMantia, A. S., Mooney, R. D., Platt, M. L., & White, L. E. (2018). Neuroscience, Sixth Edition . Oxford University Press, New York, NY. Wicks, S. R., Roehrig, C. J., & Rankin. C. H. (1996). A dynamic network simulation of the nematode tap withdrawal circuit: Predictions concerning synaptic function using behavioral criteria. Journal of Neuroscience, 16 (12):4017-4031.
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